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0593_C11_fm Page 367 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 367

O y
k

O x
kθ
n
θ

n
n r z
FIGURE 11.5.9
A free-body diagram of the pendulum rod. mg
the moment exerted on B is zero when B is in the vertical static equilibrium position. As
with the two previous examples, this system also has only one degree of freedom,
represented by the angle θ. Let G be the mass center of B. Then, the velocity and partial
velocity of G are:

v = ( ) 2 θ ˙ n θ and v = ( ) 2 n θ (11.5.12)
G
l
G
l
˙ θ
The angular velocity and partial angular velocity of B are:
˙
ωω = θn and ωω = n (11.5.13)
z ˙ θ z
Consider a free-body diagram showing the applied forces on B as in Figure 11.5.9 where
O and O represent horizontal and vertical components of the pin reaction forces. From
y
x
Eq. (11.5.7), the generalized force is:
⋅
G
F =−mg k v +( O x n + O y n ) ⋅v ˙ O − θ k n ⋅ωω θ ˙
θ
θ ˙
x
z
θ
y
(11.5.14)
l
=− ( )sin θ +− k θ
mg
2
0
Observe that because the pin O has zero velocity and as a consequence, zero partial
velocity, the pin reaction forces do not contribute to the generalized force.
11.6 Generalized Forces: Gravity and Spring Forces
The simple examples of the foregoing section demonstrate the ease of determining gen-
eralized forces. Indeed, all that is required is to project the forces and moments along the
partial velocity and partial angular velocity vectors. In this section, we will see that it is
possible to obtain general expressions for the contributions to generalized forces from
gravity and spring forces. That is, for gravity and spring forces we will see that we can
obtain their contributions to the generalized forces without computing the projections
onto the partial velocity and partial angular velocity vectors.
To this end, consider first a particle P of a mechanical system S where S has n degrees
of freedom in an inertial frame R represented by the coordinates q (r = 1,…, n). Let P have
r